I'm a young (female) professor in a (U.S.) department with a small graduate program and a research expectation. I usually teach Monday, Wednesday, and Friday, and try to save Tuesday and Thursday for research (where possible). On MWF, I'm quite good about meeting with students, even if they stop by outside of office hours (or on any day during exam weeks). My week or more homework assignments are generally due on Monday, Wednesday, or Friday (there's no good place for students to drop them off outside of class and I like discussing them as they turn them in) so I get requests by students who have put off their homework to meet on Tuesday or Thursday, even when I mention this preference in advance. I've yet to come up with a nice response that I'm satisfied both:
Reminds students I'm busy with other work (despite the fact that their tuition pays part of my salary).
that still encourages them to keep reaching out for help and encourages them to think of me as the approachable professor I try to be (at least on Monday, Wednesday, and Friday).